Effective Laguerre Asymptotics

Eﬀective Laguerre asymptotics

D. Borwein,∗ Jonathan M. Borwein† and Richard E. Crandall‡ January 18, 2007

Abstract It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Speciﬁcally, for certain ﬁxed parameter pairs (a, z) one has the large-n asymptotic √ (−a) −a/2−1/4 2 nz Ln (−z) ∼ C(a, z)n e .

We introduce a computationally motivated contour integral that allows highly efficient numerical evaluation of Ln, yet also leads to general asymptotic series over the full domain for sub-exponential behavior. We eventually lay out a fast algorithm for generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e. rigorous and explicit) error bounds on the general expansion. To this end, we avoid classical stationary-phase and steepest-descent techniques in favor of an “exp-arc” method that amounts to a natural bridge between converging series and effective asymptotics. Finally, we exhibit an absolutely convergent exp-arc series for Bessel-function evaluation as an alternative to conventional ascending-asymptotic switching.

∗Department of Mathematics University of Western Ontario, London, ONT, N6A 5B7, Canada, [email protected]. Supported in part by NSERC. †Faculty of Computer Science, Dalhousie University, Halifax NS, B3H2W5, CANADA. Email: [email protected]. J.M. Borwein’s work is funded by NSERC and the Canada Research Chair Program ‡Center for Advanced Computation, Reed College, Portland, OR 97202 USA, Email: [email protected]

1 1 The challenge of “eﬀectiveness”

The object of our interest will be the Laguerre polynomial we parameterize thus: n X n − azk L(−a)(−z) := . (1) n n − k k! k=0 Our use of negated parameters −a, −z is intentional, for convenience in our analysis and in connection with related research, as we later explain. We shall work on the diﬃcult problem of establishing asymptotics with eﬀective error bounds for the two-parameter domain

D := {(a, z) ∈ C × C : z 6∈ (−∞, 0]}.

That is, parameter a is any complex number, while z is any complex number not on the negative-closed cut (−∞, 0]. This D will turn out to be the precise domain of sub- (−a) exponential growth of Ln (−z). Herein, we say a function f(n) enjoys sub-exponential growth if log log |f(n)| ∼ δ log n, for some 0 < δ < 1, so for example f(n) = anb exp(cnδ) for real constants a, b, c with c > 0 has this growth property.1 The reason for the negative-cut exclusion on z is simple: For z negative real, the Laguerre polynomial exhibits oscillatory behavior in large n, and is not of sub-exponential growth. Note also, from the deﬁnition (1), that n − a L(−a)(0) = , (2) n n which covers the case z = 0; again, not sub-exponential growth. Now, for all (a, z) ∈ D we shall have (here and beyond we deﬁne m := n + 1, which tends to simplify notation throughout):

(−a) −1/2 Ln (−z) ∼ Sn(a, z) 1 + O(m ) , (3)

where the sub-exponential term S is √ e−z/2 e2 mz S (a, z) := √ . (4) n 2 π z1/4−a/2 m1/4+a/2 √ In such expressions, < ( mz) is taken to be pm|z| cos(θ/2) where θ := arg(z) ∈ (−π, π] (we hereby adopt the convention arg(−1) := π), and so for (a, z) ∈ D the expression (4) √ involves genuinely diverging growth in n, due to the sub-exponential exp(2 mz) factor. What we seek are eﬀective bounds, for example to replace a logical error-bounding statement for an expression E in the following way: 1 C E = O √ is replaced by E < √ for m > m0 , m m

1We use “sub-exponential” in a sense distinct from that in number theory, in which ﬁeld one typically refers to growth of order exp(d logδ n)), δ < 1.

2 with both the constant C and the threshold m0 being explicit. To do this, we eventually introduce a “Θ”-notation to simplify the nomenclature by replacing big-O when effectiveness is achieved. Thus, our chief goal in the present paper is to establish effective bounds—starting with a suitable explicit big-O constant in (3)—for the entire domain D of (a, z) pairs, even to reach further, into arbitrary asymptotic orders. Moreover, we want to effect all of this is a systematic way, so that other researchers will have a symbolic scheme for generating effective terms. To these ends, we turn next to motivational matters, and the history of Laguerre asymptotics.

1.1 Research motives A primary research motive for providing eﬀective asymptotics lies in a beautiful Laguerre series for the incomplete gamma function (see [1]), namely [14] 1 Γ(a, z) = zae−z 1 − a z + 1 1 + 2 − a z + 1 + ··· ∞ X (1 − a)n 1 = , (5) (n + 1)! (−a) (−a) n=0 Ln (−z) Ln+1 (−z) where (c)n := c(c + 1) ··· (c + n − 1) is the Pochhammer symbol. This series is valid whenever none of the Laguerre denominators has a zero. Thus an interesting sidelight is the research problem of establishing zero-free regions for Laguerre polynomials (see our Open Problems section). Indeed, one may see in this formula one good reason to adopt, as we have, Laguerre superscript (−a) and argument (−z). The asymptotic in (3) is indeed an ingredient to a separate theory of R. Crandall [9] on the large-height behavior of the incomplete gamma function, which theory in turn applies to high-precision Riemann-zeta computations. This Riemann-zeta/incomplete-gamma connection relies in turn on both imaginary heights |=(a)|, |=(z)| being very large. For example, to deal analytically-computationally with primes in the region of 1020, one might need a rigorous evaluation of something quite stultifying, say Γ 1 + 1010i, 1 + 2 · 1010i . In a word: To responsibly calculate results on prime numbers, using the analysis chain of Laguerre-polynomial → incomplete-gamma → Riemann-zeta, one must certainly know rigorous errors. It is this kind of generality√ of parameters that is incomplete in the literature. In this sense, a term O(1/ n + 1) as in (3) will not suﬃce in matters of numerical rigor unless an implied big-O constant is known. The Laguerre series for (5) suggests that the convergents of the continued fraction for the incomplete gamma function are closely related to Laguerre evaluations, and this is indeed so. In fact, another research motive we realize is that the continued-fraction

3 theory for the Γ(a, z) fractions is incomplete in the sense that the theory of S-fractions, which have been well-studied, does involve parameter restrictions. In fact, the present research began when we encountered great difficulty in estimating the convergence rate of the Γ(a, z) fraction for arbitrary a, z; this is what led to our focus on the Laguerre asymptotics. So one new research avenue is opened by this Laguerre study; namely, new results on certain continued fractions lying outside the reach of S-fractions will surely accrue. Incidentally, we are aware that sub-exponential convergence results for the general incomplete-gamma-fraction might be attainable via the very complicated, seminal work of Jacobsen and Thron [15] on oval convergence regions, but at least our effective-Laguerre approach eventually yields the desired growth properties. We should state an important caveat at this juncture: We are not intending to develop Laguerre asymptotics in order to calculate Γ(a, z) via series (5); rather, as of this writing the present effective-asymptotic analysis is the only way we know to show subexponential convergence of the Γ continued fraction for the whole domain D. If we may speak in the metamathematical vein, the intricacies of Laguerre asymptotics are not unexpected, given the corresponding intracies of complex continued fractions. There are yet other research areas that can benefit from precise Laguerre asymptotics. The celebrated hydrogen wave-functions in quantum theory involve Laguerre polynomials m Ln with n, m both positive integers [39, Ch. 4]. Beyond this, it turns out that Laguerre polynomials appear in some exactly solvable 3-body problems in quantum chemistry. In fact for the “helium-like” Hamiltonian operator for two mass-m electrons at r1, r2 and one mass-M nucleus at r3:

1 2 2 1 2 1 2 2 2 λ H = − ∇1 + ∇2 − ∇3 + mω (r13 + r23) + 2 , 2m 2M 2 r12 where rjk := rj − rk and the 2-electron repulsion is modeled here as an inverse-cube force with coupling λ, the exact wave-functions—indexed by integers n, l—are proportional to:

(a) 2 Ln 2mωr12 , with a := pl(l + 1) + 1/4 + λm. Note that for vanishing λ = 0 the Laguerre super-index a is a half-odd integer l + 1/2, but that introducing a nonzero λ breaks this symmetry so that non half-integer a become involved. In either the standard hydrogen case or the 3-body models, the asymptotic form is that of the Fejéroscillatory variety (6); yet, analyticity studies in quantum theory can involve either space, time, or both continuations into the complex plane, in which cases rigorous asymptotics may be required. Thus there is a place for Laguerre asymptotics in quantum theory. As one last example: The so-called WKB theory for approximating the remote phase of a hydrogenic wave-function is problematic, yet the oscillatory Fejér– Perron theory we next outline yields precise phases easily. One can imagine, then, a modern computationalist’s need for effective bounds in both the oscillatory and sub- exponential Laguerre cases.

4 Yet another application for growth theorems involves the Hermite polynomials, in turn expressible as n (−1/2) 2 H2n(x) = (−4) n! Ln x , n (+1/2) 2 H2n+1(x) = 2(−4) n! x Ln x , and appearing at many important theoretical junctures in quantum theory. As with the aforementioned atomic wavefunctions, it is important to know the asymptotic behavior of the Hermite class. For one thing, space-time and space-energy propagators for the Schr¨odinger theory sometimes demand asymptotic analysis in regard to eigenvalue estimates. In any case, our present results do apply to the Hermite class, via the above √ √ √ even-odd identities (one simply asigns x := −z with our branch rule −ρ := i ρ for positive real ρ). One interesting work—whose methods are distinct from our present ones—involves explicit bounds on Hermite oscillations [12]. And, for asymptotic analysis, there may be some hope in the interesting L´opez–Temme expansion:

n a−z−1 X ckHn−k L(−a)(−z) = (−x)n 2x , n xk(n − k)! k=0 p where x := −z − (1 − a)/2 and the ck are generated by a certain 4-th order recurrence relation [18]. Strikingly nonstandard as this representation may be, it is neverthless valid for all parameter pairs (a, z) ∈ C × C.

1.2 Historical results Laguerre asymptotics have long been established for certain restricted domains, and usu- ally with noneﬀective asymptotics.2 For example, in 1909 Fej´erestablished that for z on the open cut (−∞, 0) and any real a, one has [30, Theorem 8.22.1]:

e−z/2 √ L(−a)(−z) = √ cos 2 −mz + aπ/2 − π/4 + O m−a/2−3/4 , (6) n π(−z)1/4−a/2 m1/4+a/2 where we again use index m := n+1, which slightly alters the coefficients in such classical expansions, said coefficients being for powers n−k/2 but we are now using powers m−k/2. By 1921 Perron [28] had generalized the Fejérseries to arbitrary orders, then for z 6∈ (−∞, 0] established a series consistent with (3) and (4), in essentially the form: [30, Theorem 8.22.3]

N−1 ! X Ck L(−a)(−z) = S (a, z) + O m−N/2 , (7) n n mk/2 k=0 although this was for real a and so not for our general parameter domain D. Note that C0 = 1, consistent with our (3) and (4); however, one should take care that because we

2Some researchers use the term “realistic error bound” for big-O terms that have explicit structure. We prefer “eﬀective bound,” and when an expansion is bestowed with such a bound, we may say “eﬀective expansion.”

5 are using index m := n + 1, the coefficients Ck in the above formula differ slightly from the historical ones. Various of the Perron generalizations can be derived, for example, from the Le Roy formula, valid for <(a) > 1 and z negative real, Z ∞ (−a) a/2 −z 1 n−a/2 −t √ Ln (−z) = (−z) e t e J−a(2 −zt) dt, n! 0 which formula is more than a century old (see [30, ch. 5.4, p. 99], where extension to more general parameters is discussed). That is, one may use the classical Bessel-expansion theory to obtain a Laguerre expansion. The complications attendant on this approach are a good reason to adopt, as we shall, a more universal contour integral representation. It is not hard to see how the interplay between domain of z and the asymptotic character of (6) or (7) works. If we naively take 0 6= z 6∈ (−∞, 0) in the Fejér form (6), the cosine term has a sub-exponentially growing part (1/2) exp(2pm|z| cos(θ/2))—so even the extra factor of 1/2 contained in the Sn definition is explained in this way. Conversely, one can imagine moving from the sub-exponential form (7) to the oscillatory form (6) as z moves onto the cut (−∞, 0) from outside it, provided one remembers the mnemonic that √ there is in (3) or (7) a “buried” term like Sn but having exp(−2 mz) in the numerator, such a term being sub-exponentially small. All of these heuristics and mnemonics can be √ remembered perhaps best by imagining a cosh(2 mz) term in the definition of Sn, with the denominator’s ‘2’ removed. It will turn out that our rather involved effective-error √ analysis moves most cleanly, though, with (a, z) ∈ D and the exp(2 mz) term intact. Certain asymptotic properties may be gleaned from the formal generating function

∞ zt X (−a) n a−1 1−t u(t) := Ln (−z)t = (1 − t) e , (8) n=0 as can be derived from standard ordinary differential equation theory by applying the P n recurrence (10) in the next section to the generating function n rnt . A more modern literature treatment that is again consistent with the heuristic (3)–(4), is given by Wipitski (−a) in [37], where (8) is invoked to yield a contour integral for Ln (−z). Then a stationary- phase approach yields precisely the correct asymptotic, at least for certain subregions of D. It should be noted however that Wipitski’s treatment is both elegant and non-rigorous, intended mostly as a computational guide. And once again, there is no estimate given on √ the O(1/ m) correction in (3). In fact, our present approach involves the avoidance of stationary-phase approach per se, in favor of an exponential-arcsin series development, as described in Section 5. There is an interesting anecdote that reveals the difficulty inherent in Laguerre asymptotics. Namely, W. Van Assche in an interesting 1985 paper [34] used the expansion (7) for work on zero-distributions, only to find by 2001 that the C1 term in that 1985 paper had been calculated incorrectly. The amended series is given in his correction note [35] as √ e−z/2 e2 nz 3 − 12a2 + 24(1 − a)z + 4z2 1 1 L(−a)(−z) ∼ √ . 1 + √ √ + O , n 2 π z1/4−a/2 n1/4+a/2 48 z n n

6 or in our own notation with m := n + 1,

3 − 12a2 − 24(1 + a)z + 4z2 1 1 ∼ S (a, z) 1 + √ √ + O . (9) n 48 z m m √ Note the slight alteration used to obtain our C / m term. Van Assche credits T. Müller 1 √ and F. Olver for aid in working out the correct O(1/ n) component.3 This story suggests that even a low-order asymptotic development is nontrivial. Hence, we should anticipate the effective error-bounding project on which we embark to be at least as nontrivial. At any rate, along the way we shall streamline a symbolic-generation process for obtaining the Ck, and in this way shall give an algorithm for extending (9) to higher orders and provide effective bounds for the corresponding error terms Ek. Absent the problem of incomplete domains (incomplete for our modern purposes), the classical authors certainly knew in principle how to establish effective error bounds. The excellent treatment of effectiveness for Laplace’s method of steepest descent in [22] is a shining example. Even more illuminating is Olver’s paper [21], which explains effective bounding and shows how unwieldy rigorous bounds can be. However, efficient algorithms for generating explicit effective big-O constants have only become practicable in recent times. We freely admit that here-and-now in the era of modern symbolic processing—and in spite of the complications into which we soon shall delve—the generation of explicit asymptotic terms and their associated errors is easier today than at any time in the history of the subject. We note that many alternative Laguerre studies abound. One modern thrust—which we do not address in the present paper—involves asymptotic behavior when the subindex n is linked either to a or z, or both. For example, the paper [17] provides a large-n asymptotic (a) (nx) expansion for Ln (nx). Another treatment is [29], in which the author handles Ln (ny). (an) In [8] the authors analyze cases of Ln for which an >> n. One hopes that our present methods for effective bounding can be applied to such variants. Other important “linked” cases include the exponential monomial and the partial-exponential sum, respectively:

n (−1)n X zk L(−n)(z) = zn; L(−n−1)(z) = (−1)n . n n! n k! k=0 The paper [16] discusses such generalizations. An important point here is that any asymptotic theory with n ≈ a would have to take into account these useful special cases. On (µ) the subject of rigorous bounds, there is the interesting treatment [19] in which Lν (x) is given upper (and lower!) bounds, for real x and <(µ) > −1. One might call such results eﬀective error bounds of zero-th order; in any case, they do not help the present treatment directly because of their parameter-domain restrictions.

3 Accordingly, we hereby name the polynomial C1(a, z) the Perron–van Assche–M¨uller–Olver, or “PAMO” coeﬃcient.

7 1.3 Further relations for Laguerre polynomials Here we give further relations for Laguerre polynomials, including some relations we do not use in the present treatment, but which may nevertheless be helpful in future research on the problem of eﬀective bounds. (−a) (−a) We record what might be called initial values: L0 (−z) = 1, and L−1 (−z) := 0. We then note that the iteration (n ≥ 1) A B r = 2 − r − 1 − r , (10) n n n−1 n n−2 with initial values r−1 = 0, r0 := 1, thus has an exact solution in the form

(−a) rn = Ln (−z), (11) where z := B − A, a := B − 1, is the parameter transformation that moves us between equivalent pairs (a, z) and (A, B). Thus, the complex domain in question for (A, B) pairs can be inferred from our caveat (a, z) ∈ D. The reason we have emphasized here the rn-iteration (10) itself is that direct analysis of second-order recurrences have, on other problems, yielded strong results [5, 6, 7]. A “discrete” approach that attempts alternative asymptotic expressions for the rn is therefore promising. Perhaps just as promising for growth analysis is the Laguerre differential equation, which in our present parameterization reads ∂2L ∂L −z + (a − 1 − z) + nL = 0, (12) ∂z2 ∂z (−a) where L := Ln (−z). In fact, various sharp results on Laguerre asymptotics have emerged from differential theory—there is the classical work of Erdélyiand Olver, plus modern work on combinations of differential, discrete, and saddle-point theory [11][13]. Next, for some hypergeometric connections, there is a hypergeometric form n − a L(−a)(−z) = F (−n, 1 − a; −z) , (13) n n 1 1 where 1F1 is also known as the Kummer (confluent) hypergeometric function. In references such as [1, §13.5.14] the Kummer function 1F1 asymptotics are consistent with aforementioned Laguerre forms, but again such formulae are established for restricted subregions of D and without effective bounds. When n is small (in some sense not made rigorous here), and say |z| |a|, an approximation follows from the representation (8), as

n j n − a X (−n)j (−z) L(−a)(−z) = . n n (1 − a) j! j=0 j

8 Because n n (−z) n Γ(1 − a) (−z) c1(n) c2(n) = (−z) ∼ n 1 + + 2 + ... , (1 − a)n Γ(1 − a + n) (1 − a) 1 − a (1 − a)

1 where the cj(n) are polynomials in n with c1(n) = − 2 n(n − 1), this gives a first order approximation n − a z n L(−a)(−z) ∼ 1 + . n n 1 − a More terms in such an expansion can be obtained via simple operations.4 We have not been rigorous about these small-n approximations, though we believe further research should certainly yield effective bounds. It is a fascinating phenomenon that the character of the above small-n expansion changes dramatically for large n—the behavior switches from a kind of modified power law to sub-exponential growth.

1.4 An outline of what follows In Section 2 we obtain a tripartite contour representation for the generalized Laguerre polynomials and explore to which parts contribute predominantly. In Section 3 we provide eﬀect bounds for each of three contour integrals we call c1, d1, e1. We summarize the status of contour integration in Section 3.5. In Section 4 we attack a dominant term c0 (of c1), and reduce the eﬀective-bounds problem to the study of a Bessel-class integral

Z π/2 I(p, q) := e−iqωep cos ω dω. −π/2

In Section 5 we conquer the asymptotics of I by exploiting the power series in x of exp(τ arcsin(x)), calling this an “exp-arc” series. Section 6 then harvests our crop: The- orem 8 gives an eﬀective expansion for the generalized Laguerre polynomials in the sub- exponential region. Section 6.2 then provides an algorithm for the purpose while Section 6.3 shows the method in numerical mode. In Section 7 we brieﬂy consider the oscillatory domain before applying our results to the Bessel functions In and Jn of integral order. Finally, in Section 8 we visit or revisit some of the open problems thrown up by our study.

2 Contour representation

In this section we develop a highly eﬃcient—both numerically and analytically—contour- (−a) integral representation for Ln (−z). First we indicate how experimental mathematics was employed to work out a good contour itself, then we proceed to provide effective bounds on segments of the contour, whence extracting a primary term that is sub-exponentially larger than the other terms.

4The authors are indebted to N. Temme for these small-n approximations.

9 2.1 Development of a “keyhole” contour A well known contour integral has contour, Γ, encircling s = 1 + 0i but avoiding the branch cut (−∞, 0]:

−z Z −n−1 (−a) e −1−a 1 zs Ln (−z) = s 1 − e ds. (14) 2πi Γ s This representation holds for all pairs (a, z) ∈ C × C, n a non-negative integer, and may be proven to agree with the polynomial definition (1) via simple residue arithmetic, or via a Fourier transform of the generating function (8).5 However—and this is important—we found via experimental mathematical techniques that a certain kind of contour allows very accurate, efficient, and well behaved numerical Laguerre evaluations. Take z 6= 0, m := n+1 and assume r := pm/z has |r| > 1/2. Then, use a circular contour centered at s = 1/2 with radius |r|. This contour will encompass s = 1, so the remaining requirement is to avoid the cut s ∈ (−∞, 0]. But this can be done by cutting out a “wedge” from the negative-real arc of the circle, with the wedge’s apex at 1/2 = 0i. We actually tried such schemes with high-precision integration, to settle finally on the contour of Figure 1, where the aforementioned wedge has become a “keyhole” pattern consisting of cut-run D1 and small, origin-centered circle E1 of radius 1/2.

So, adopting constraints and nomenclature:

z 6= 0, θ := arg(z), ω± := ±π + θ/2, m := n + 1, r := pm/z := pm/|z|e−iθ/2,R := |r| > 1/2, but no other constraints, we have the following representation for L:

(−a) Ln (−z) = c1 + d1 + e1, (15)

where c1, d1, e1 are the respective contributions from contour C1, cut-discontinuity D1, and contour E1 from Figure 1. Exact formulae for said contributions are

Z ω+ √ 1 −a −z/2 −iω 2 mz cos ω c1 = r e Hm(a, z, e )e dω, (16) 2π ω−

−z Z R−1/2 −m e −1−a 1 −zT d1 = sin(πa) T 1 + e dT, (17) π 1/2 T

−z Z π e −iω1+a −iω−m iω+ z eiω e1 = − 2e 1 − 2e e 2 dω, (18) 4π −π 5The contour representation (14) can easily be continued to non-integer n, with care taken on the (−n−1)-th power, but our present treatment will only use nonnegative integer n.

10 (−a) Figure 1: A numerically eﬃcient “keyhole contour” for Laguerre evaluations Ln (−z), valid for all complex a, z with z 6= 0 and n + 1 > |z|/4. Wedges with center-1/2—as pictured at right— are experimentally accurate, leading to a “keyhole” deformation avoiding the cut s ∈ (−∞, 0]. It turns that for any pair (a, z) ∈ D, the main arc C1 gives the predominant contribution for large n, the D1,E1 components being sub-exponentially minuscule.

and when m > <(a) we may write this last contribution by shrinking down the radius-1/2 contour segment to embrace the cut (−1/2, 0], as

−z Z 1/2 −m e −1−a 1 −zT e1 = sin(πa) T 1 + e dT. (19) π 0 T

For the c1 contribution above, we have used the function H deﬁned by v −1−a h v im H (a, z, v) := va 1 + F , (20) m 2r r with 1 + t/2 F (t) := e−t, (21) 1 − t/2

which for t small is: 1 + t3/12 + t5/80 + ··· = 1 + O(t3).

Note. The form e1 given by (18) is more general than (19) as is exempliﬁed by the case (−7/2) a := 7/2, z := −1, m := 3 for which L2 (1) = 31/8 with the (18) form contributing correctly, while the integral (19) does not even exist. Another example is a := 5, z :=

11 −1, m := 3 for which (19) has a vanishing prefactor sin(πa), yet e1 ≈ 9.513 ... as provided by (18) is correct.

In any event, e1, like d1, is sub-exponentially small relative to c1, for large n and (a, z) ∈ D. It is to be stressed that decomposition (15) holds for all (a, z) ∈ C × C, z 6= 0, as long as m := n+1 > |z|/4. We remind ourselves of (2) for z = 0. This means that such contour calculus applies to both oscillatory (Fej´er)cases, where z is negative real, and sub-exponential (Perron) cases for the stated parameters. We shall soon be restricting (a, z) to the domain D, which restriction will induce sub-exponential growth always, as in (3), with eﬀective bounds attainable for various contour segments.

2.2 Numerical assessment of the contour

The triumvirate c1, d1, e1 of integrals is suitable for accurate Laguerre computations, which computations do show that integrals d1, e1 tend to be sub-exponentially small relative to c1. And this, of course, is our motive for identifying the contour terms in such a way. An example computation would be, from the deﬁning series (1) with (a, z) := (−i, −1 − i): 137 53 L(i)(1 + i) = − + i 8 288 45 which to 20-place accuracy is

≈ −0.47569444444444444444 + 1.17777777777777777777i,

whereas the respective contributions from the C1,D1,E1 segments of the keyhole contour of Figure 1 turn out to be

c1 ≈ −0.44406762576110996056 + 0.81722282272705891241i,

d1 ≈ −0.03169598827425878852 + 0.36065591639721657587i,

e1 ≈ 0.00006916959092430464 − 0.00010096134649771051i, (i) with the sum c1 + d1 + e1 giving L8 (1 + i) to the implied precision. Remarkably, this “keyhole-contour” approach has the additional, unexpected feature (a) that for some parameter regions the contour evaluation of Ln (−z) is actually faster then direct summation of the deﬁning series (1). A typical example is: For the 13-digit evaluation

(25i/2) 1056 L50000 (−30 + 3i) ≈ (0.9275136583293 + 1.7406691595239i) × 10

the deﬁning series (1) was, in our trials, three-times slower than the integral c1 (the other two contour segments are well below the 13-digit signiﬁcance).6

6Of course, series acceleration as in [3] would give the direct series a “leg up.” Still, it is remarkable that contour integration would even be competitive. For the record, we compared Mathematica’s numerical integration to its own LaguerreL[ ] function.

12 3 Eﬀective bounds on the contour components 3.1 Theta-calculus We now introduce a notation useful in eﬀective-error analysis. When two functions enjoy

|f(z)| ≤ |g(z)| over some domain z ∈ X , we shall say that

f(z) = Θ(g(z)) on said domain. Thus, the Θ-notation is an effective replacement for big-O notation. For example, for z on the complex unit circle, we have z = Θ(2), z + 1/2 even though this fails for some z off the circle. This “theta-algebra” obeys certain relations, such as this for any finite function evaluations fk:

|f1 + f2 + ... | ≤ Θ(f1) + Θ(f2) + ..., from the classical triangle inequality. (Note however that one may not always write the right-hand side as Θ(f1 + f2 + ... ) because of possible cancelation.) Also easy is the multiplicative relation for arbitrary complex multipliers α:

αΘ(f(z)) = Θ(αf(z)).

When an expression g is unwieldy, we allow ourselves the luxury of denoting Θ(g) by Θ: g, so that a long formula g can run arbitrarily to the right of the colon.

3.2 Eﬀective bound for e1

First we address the integral e1 as given in relation (18). We need some preliminary lemmas, starting with a collection of polynomial estimates to transcendental functions.

Lemma 1 The following inequalities hold: 1. On ω ∈ [−π, π] we have log 9 log(5 − 4 cos ω) ≥ ω2. π2 2. On ω ∈ [0, 1/2] we have 4 log(1 + ω) ≥ ω. 5 √ 3. On ω ∈ [0, 1/ 2] we have π arcsin ω ≤ √ ω. 8

13 4. On ω ∈ [−π/2, π/2] we have 4 cos ω ≤ 1 − ω2. π2

Proof. All four are straight-forward calculus exercises. We illustrate with a proof the ﬁrst which is slightly more subtle. Observe that 1. is equivalent to log 9 log(1 + 8 sin2(x/2)) ≥ x2. π2 Since sin(t) ≥ 2t/π when 0 ≤ t ≤ π/2, we have

log(1 + 8 sin2(x/2)) ≥ log(1 + 8x2/π2).

Putting u := 2x2/π2, we see that it suﬃces to prove

log(1 + 4u) ≥ u log 3 for 0 ≤ u ≤ 2, which follows by simple calculus. QED

Lemma 2 Consider integrals of error-function class, speciﬁcally, for nonnegative real parameter µ, Z ∞ 2µ 2αx−βx2 Vµ(α, β, γ) := x e dx, γ where each of α, β, γ is real, with α, β > 0. Then we always have the bound

rπ α2 V (α, β, γ) = Θ : e β . 0 β

If in addition γ > 2α/β, we also have

1 Γ(µ + 1/2) V (α, β, γ) = Θ : ; µ 2 (β − 2α/γ)µ+1/2

ﬁnally, if γ > 4α/β we have

Γ(µ + 1/2) V (α, β, γ) = Θ : 2µ−1/2 , µ βµ+1/2

Remark: The last two bounds we shall not use until a later section, but it is convenient to dispense with all these error-function results now. Proof. Completing the exponent’s square gives the ﬁrst bound easily, since Z ∞ α2/β −β(x−α/β)2 V0 = e e dx γ

14 with the integral here being Θ pπ/β . For the second bound, it is elementary that for x ≥ γ one has 2αx − βx2 ≤ x2(2α/γ − β), so that Z ∞ 2µ −(β−2α/γ)x2 Vµ ≤ x e dx 0 and the resulting bound follows. The ﬁnal bound follows immediately from the previous bound, because γ > 4α/β implies β − 2α/γ > β/2. QED

Theorem 1 The contour contribution e1 deﬁned by (18), under conditions (a, z) ∈ D and m suﬃciently large in the explicit sense

m := n + 1 > m0 := |z|/4,

2 m > m1 := 5 |<(z)| + (|=(a)| + |=(z)|/2) , is bounded as 1 e = Θ : e−z/22<(a)+3 √ . 1 m Moreover, under alternative conditions (a, z) ∈ D and

m > m0, m > m2 := <(a), 5 m > m := − |z| cos θ, 3 4 we have a bound e−z 2<(a)−m e = Θ : sin(πa) 1 π m − <(a)

Proof. First, in (18) the term (1 − 2e−iω)−m has absolute value (5 − 4 cos ω)−m/2. So we can write, recalling θ := arg(z) ∈ (−π, π],

−z/2 Z π e <(a) ω=(a) −(m/2) log(5−4 cos ω) |z|(cos(ω+θ)−cos θ)/2 e1 = Θ : 2 e e e dω. 2π −π Now the overall exponent here can be written m 1 ω=(a) − log(5 − 4 cos ω) − <(z) sin2(ω/2) − =(z) sin ω. 2 2 Then, via Lemma 1 we can give an upper bound for this real expression when ω ≥ 0, namely log 9 (|=(a)| + |=(z)|/2)ω − mω2 − <(z) sin2(ω/2). 2π2 Now deﬁne parameter α := |=(a)|/2 + |=(z)|/4,

15 and if <(z) < 0 deﬁne m log 9 β := + <(z)/4, 2π2 but if <(z) ≥ 0 we drop the term <(z)/4 in the deﬁnition. With this nomenclature, we can write −z/2 Z π e <(a) 2αω−βω2 e1 := Θ : 2 2 e dω 2π 0 e−z/2 = Θ : 2<(a)V (α, β, 0). π 0

Under the stated condition m > m1, we have |<(z)| < m/5 and so log 9 1 m β > − m > . 2π2 20 20

Also from the same fact m > m1 we have m α2 < < β. 20 whence Lemma 2 gives −z/2 e <(a) e e1 = Θ : √ 2 , π pm/20 and the desired result follows. For the second set of conditions on m, the fact of m > m2 means that (18) and (19) are equivalent, and Lemma 1 allows:

−z Z 1/2 e m−1−<(a) −zT −4mT/5 e1 = Θ : sin(πa) T e dT. π 0

<(a)−m Since the integral is bounded for m > m3 by 2 /(m − <(a)), the desired bound follows. QED

3.3 Eﬀective bound for d1 Again we need an opening lemma:

Lemma 3 Consider integrals of the incomplete–Bessel class, speciﬁcally

Z 1 −αx− β dx W (α, β) := e x , 0 x where each of α, β is real, with β > 0. If β > α we have a bound

1 rπ W = Θ : e−β−α . 2 β

16 Proof. We write Z 1 Z 1 βx−αx −βx− β dx β−α −βx− β dx W (α, β) = e e x ≤ e e x 0 x 0 x ∞ ∞ Z Z 2 = eβ−α e−2β cosh t dt ≤ e−β−α e−βt dt, 0 0 which proves the theorem. (Incidentally, the integral with cosh in the exponent is the modiﬁed-bessel evaluation K0(2β) which is known to have the necessary bound [2, Lemma 1].) QED

Theorem 2 The contour contribution d1 deﬁned by (17) can be bounded under conditions (a, z) ∈ D and m > m4 := 4|z|, as −z/2 √ e |<(a)|/2−1/4 −|<(a)|/2−1/4 −2 m|z| cos2 θ d = Θ : √ m |z| sin(πa)e 2 . 1 π

p Proof. From (17) we have (noting that m > m4 means R := m/|z| > 2,

−z/2 Z R e −1−a −zt+m(log(1−1/(2t))−log(1+1/(2t))) d1 = sin(πa) (t − 1/2) e dt. π 1 On the interval t ∈ (1,R) we have

(t − 1/2)−1−a = Θ : (t − 1/2)−1(t − 1/2)−<(a)

2 |<(a)| which, since R > 2, is Θ : t R . Thus −z/2 Z 1 √ √ e |<(a)| dτ −τ m|z| cos θ− m|z|/τ d1 = Θ : sin(πa)2R e . π 1/R τ The desired bound then follows immediately from Lemma 3. QED

3.4 Rigorous estimates on c1

Having dispensed with d1, e1, we next show that the integration limits ω−, ω+ on the c1 contribution can be changed—with only a sub-exponentially small error penalty—to −π/2, π/2 respectively, as we now establish. We start with

Lemma 4 For any v on the unit circle {eiφ : φ ∈ (−π, π]}, any complex a, and any real R > (1 + |a|)/2 we have

v −1−a 1 1 + ≤ . 2R 1+|a| 1 − 2R

17 Proof. From the binomial theorem, v −1−a −1 − a v (−1 − a)(−2 − a) v 2 1 + = 1 + + + .... 2R 1! 2R 2! 2R The right-hand side is bounded in absolute value by 1 1 2 1 + (1 + |a|) + (1 + |a|)2 + ... 2R 2R 1 = . 1+|a| 1 − 2R QED Lemma 5 Let v be on the unit circle as in Lemma 4, let R > 1 be real, and let m be a positive integer. For the function F appearing in (21), we have the bound

v m 1 m F ≤ e 6 R3 . R Proof. From the deﬁnition (21) we have

m 1 m (1+(3/5)/(2R)2+(3/7)/(2R)4+... ) F (v/R) = e 12 R3 . For R := 1 the parenthetical inﬁnite sum is no larger than 1.18... and is monotonic decreasing in R. QED

Lemma 6 Let v be on the unit circle as in Lemma 4 and deﬁne for nonnegative integer m v −1−a v m K := 1 + F . 2R R p 2 For the assignments (a, z) ∈ C×C, z 6= 0,R := m/|z| > 1, m > m5 := |z|(1+|a|+|z|/2) . Then |K| ≤ 2.

Proof. From Lemmas 4, 5 we have

1 1 |z| |K| ≤ e 6 1+|a|+|z|/2 . 1+|a| 1 − 2(1+|a|+|z|/2) The fact that the right-hand side is Θ(2) follows from the observation that the function

1 1 x 3 2Q+x Q e 1 − 2Q+x for Q ≥ 1, x ∈ [0, ∞) is itself Θ(2), and this in turn follows easily from the substitution y := x/(2Q + x), for which the function to be bounded is g(y) := (2/(1 + y))ey/3 on y ∈ [0, 1/2]— use of the derivative dg/dy settles this. QED

These lemmas in turn allow us to contract the range on the c1 contour integral:

18 Theorem 3 Decompose c1 as deﬁned by (16) into two terms,

c1 := c0 + c2, with c0 involving the integral’s range contracted to [−π/2, π/2], namely

Z π/2 √ 1 −a −z/2 −iω 2 mz cos ω c0 := r e Hm(a, z, e )e dω, 2π −π/2

Then under conditions (a, z) ∈ D and

2 m > m5 := |z|(1 + |a| + |z|/2) we have a bound on the remaining integral

−a −z/2 3 π|=(a)| c2 = Θ : r e e 2 .

Proof. It is evident that c2 is obtained from the deﬁnition (16) but with the integral replaced according to ( ) Z ω+ Z −π/2 Z ω+ → + . ω− ω− π/2 √ Over these domains of integration, however, we have e2 mz cos ω = Θ(1), and from Lemmas 4, 5, 6 we have ( ) Z −π/2 Z ω+ 1 −a −z/2 −iωa c2 = Θ : r e + 2 e dω, 2π ω− π/2 and as the total support of the integrals cannot exceed π, the desired c2 bound follows easily. QED

(−a) 3.5 Summary of the contour decomposition for Ln (−z) The above machinations lead to the main result of the present section, namely a formula that decomposes the Laguerre evaluation, as in

Theorem 4 (Contour decomposition) Let (a, z) ∈ C × C be an arbitrary parameter pair with z 6= 0 (with z = 0 cases resolved exactly by (2)). If

m := n + 1 = m0 > |z|/4, then the contour decomposition

(−a) Ln (−z) = c0 + E

19 holds, with c0, c2 as deﬁned in Theorem 3 and E := c2 + d1 + e1. If an appropriate union of conditions on m, a, z across Theorems 1, 2, 3 holds, then we can write

(−a) Ln (−z) = c0 + Sn(a, z)E1, where E1 is sub-exponentially small, in the sense that for any positive the large-n behavior is √ −(2−) m|z| cos θ E1 = O e 2 . Moreover, an eﬀective big-O constant is available (the proof exhibits explicit forms).

Proof. The contour calculus is valid for all complex pairs (a, z) when z 6= 0,R := pm/|z| > 1/2, which conditions assure that the point 1 + 0i is contained in the contour. (−a) It remains to analyze Ln (−z) = c0 + Sn(a, z) E1. Consider, then, an appropriate union of conditions from the cited theorems, say

(a, z) ∈ D,

m > max(m0, m1, m2, m3, m4, m5) in which case we have, from Theorems 1, 2, 3 (in that respective order of Θ terms):

−z/2 1+<(a)−m √ e 2 1/4−a/2 1/4+a/2 −2 m|z| cos θ E = Θ : √ sin(πa) z m e 2 1 π m − <(a)

(<(a)+|<(a)|)/2 √ m 3 π|=(a)| −2 m|z| cos2 θ +cos θ +Θ : 2 e 2 e ( 2 2 ) |z| √ √ 1/4 3 π|=(a)| −2 m|z| cos θ +Θ : 2 π |mz| e 2 e 2 . √ −(2−) m|z| cos(θ/2) This explicit bounding of the error term E1 proves the O e statement of the theorem, while for any choice of ε an eﬀective big-O constant can be read oﬀ at will. QED

Theorem 4 suggests—and we shall prove—that the c0 term deﬁned in Theorem 3 can be given a Perron-like series, essentially as in (7). This will establish that the precise (a, z)-parameter domain of large-n sub-exponential growth will be D. Thus our remaining tasks are to provide:

1. A symbolic algorithm for the asymptotic coeﬃcients Ck in (7), and

2. An eﬀective bound for the EN term. Moreover, we wish to achieve these things over the whole domain of sub-exponential Laguerre growth, namely for all (a, z) ∈ D.

20 4 Eﬀective expansion for the H-kernel

(a) In the sense of Theorem 4, the dominant contribution to Ln (−z) for (a, z) ∈ D, with r := pm/z, |r| > 1/2, is

Z π/2 √ 1 −a −z/2 −iω 2 mz cos ω c0 := r e Hm a, z, e e dω, (22) 2π −π/2 with the integration kernel H deﬁned, see (20) and (21), as

−1−a v m a v 1 + 2r −mv/r Hm(a, z, v) := v 1 + v e , (23) 2r 1 − 2r where in the integral we assign v := e−iω. We need to obtain the growth properties of H. This we do in the next three subsections. We suspect there is a more standard combinatorial route to Theorem 5 of Section 4.3, but it has so far eluded us.

4.1 Exponential form for H Lemma 7 For |v| = 1 and m > |z|/4, the H-kernel can be cast in the exponential form   X ak 1  H := va exp , (24) m k mk/2 k≥1  where √ √ v z k k v z k+2 a := (1 + a)(−1)k + 1 − (−1)k . (25) k 2 k + 2 2 Moreover, we have the general coeﬃcient bound √ z k |a | ≤ (1 + |a| + |z|/2). (26) k 2

Proof. (23) can be recast, with ρ := v/(2r), as

a Hm := v exp {−(1 + a) log(1 + ρ) − 2mρ + m (log(1 + ρ) − log(1 − ρ))} . (27)

Being as |r| > 1/2 and |v| = 1, the logarithmic series converge absolutely and we have   " √ k √ 2k+1 # X (1 + a)(−1)k v z 2 v z  H := va exp gk + g2k−2 , (28) m k 2 2k + 1 2 k≥1  √ where g := 1/ m, and the precise form (25) for the ak follows immediately. The given bound on |ak| is also immediate from (25). QED

21 4.2 Exponentiation of series Though Lemma 7 is progress, we still need to exponentiate a series, in the sense that we want to know, for the following expansion, given the sequence (ak),   X ak  X exp xk =: A xh, k h k≥1  h≥0 how the Ah depend on the ak. The combinatorial answer is

h X 1 A = G (j;~a), h j! h j=0 where X ah1 ··· ahj Gh(j;~a) := , h1 ··· hj h1+...+hj =h with the understanding Gh(0,~a) := δ0h and that such combinatorial sums involve positive integer indices hi. One result that can prove useful for such combinatorics is a bound on a simple instance of the G sum:

Lemma 8 If all coeﬃcients ak are equal 1 then, for j > 0, X 1 Gh(j;~1) = h1 ··· hj h1+...+hj =h 1 = Θ : (2H )j−1 , h h−j+1 1 = Θ : (2γ + 2 log h)j−1 . h Here, Hq := 1 + 1/2 + ··· + 1/q is the q-th Harmonic number, H0 := 0, and γ is the Euler constant.

Remark. It turns out that G here enjoys a closed form of sorts, namely j! G (j;~1) = (−1)h−jS(j), h h! h where S denotes the Stirling number of the ﬁrst kind, normalized via x(x − 1) ··· (x − h + Ph (j) j 1) =: j=0 Sh x . So the eﬀective of our lemma is a rigorous bound on the growth of Stirling numbers; see [1, 24.1.3,III] and [31] for research on Stirling asymptotics.

Proof. The ﬁrst Θ-estimate arises by induction. For notational convenience we omit the vector ~1 and just use the symbol Gh(j). Note GN (1) = 1/N and

N−1 X 1 2 G (2) = = H . N j(N − j) N N−1 j=1

22 Generally we have N−J+1 X 1 G (J) = G (J − 1). N j N−j j=1

1 j−1 Now, assume by induction that Gh(j) = Θ : h (2Hh−j+1) , holds for all j < J. Then

N−j+1 J−2 J−2 X 2 HN−J−j+2 G(N,J) ≤ j(N − J) j=1

N−J+1 2J−2 X 1 1 ≤ HJ−2 + . N N−J+1 j N − j j=1

Now the parenthetical term is HN−J+1 + HN−1 − HJ which, because Ha − Hb ≤ Ha−b for any positive integer indices a > b, is bounded above by 2HN−J+1, which proves the ﬁrst Θ-bound of the theorem. For the second θ-bound it suﬃces to show that Hn−1 > γ + log(n), since Hj is increas- ing. We set sn := Hn−1 − log(n). Then, s1 = 0 and for n > 0

1 1 s − s = − log 1 + > 0, n+1 n n n and so sn increases and, by deﬁnition, tends to γ as n → ∞. QED

Though we do not use Lemma 8 directly in what follows, it is useful in proving con- P h vergence for various sums Ahx , and may matter in future research along our lines.

Lemma 9 Let y ≥ 1 and x ∈ (−1, 1) be real. Then in the expansion    X xk  X exp y =: Y xh k h  k≥1  h≥0 the coeﬃcients Yh enjoy the bound

h Yh = Θ y .

Proof. The left-hand side is exp(−y log(1 − x)) = (1 − x)−y whose binomial expansion has h-th coeﬃcient (h ≥ 1) equal to

y(y + 1) ··· (y + h − 1) 1 (1 + 1/y) 1 + (h − 1)/y ≤ yh ··· ≤ yh. h! 1 2 h

Finally, Y0 = 1 ≤ 1. QED

23 4.3 Eﬀective expansions of exponentiated series We are now in a position to contemplate an eﬀective expansion for an exponentiated series, starting with

~ Lemma 10 Assume complex vector b of deﬁning coeﬃcients bk bounded as

k |bk| ≤ cd for positive real c, d with c ≥ 1. Let real x satisfy |x| < 1/(2cd). Then for any order N ≥ 0 we have an eﬀective expansion

  N−1 X bk  X exp xk = B xh + Θ 2cN dN xN , k h k≥1  h=0 where h ~ X Gh(j; b) B = h j! j=0 are the usual coeﬃcients of the full formal exponentiation.

n o P bk k Proof. Denoting f(x) := exp k≥1 k x we have

N−1 X h f(x) = Bhx + TN . h=0

h |Bh| ≤ (cd) .

Therefore X cN dN xN |T | ≤ (cd)hxh = = Θ 2(cdx)N . N 1 − cdx h≥N QED

Finally we arrive at a general expansion—with eﬀective remainder—for the H-kernel:

24 Theorem 5 (Eﬀective expansion for H.) For general complex (a, z) ∈ C × C, assume 2 m > m5 := |z|(1 + |a| + |z|/2) and |v| = 1. Then for any expansion order N ≥ 0 we have N−1 ! X Ah m5 N/2 H (a, z, v) = va + Θ : 2 , m mh/2 4m h=0 where h X Gh(j;~a) A := , h j! j=0

X ah1 ··· ahj Gh(j;~a) := , h1 ··· hj h1+...+hj =h with the deﬁning coeﬃcients ak given in (25).

Proof. The result follows immediately from Lemma 10, on assigning ~b = ~a, with x := √ 1/ m, c := 1 + |a| + |z|/2, d := (1/2)p|z|. QED

Note that Theorem 5 in the instance N = 0 implies our previous Lemma 6. It is 2 interesting and suggestive that the threshold m5 := |z|(1 + |a| + |z|/2) appears in both theorem and lemma rather naturally.

4.4 Eﬀective integral form for c0 (−a) To obtain a useful form for c0, the dominant component of Ln (−z), we use Theorem 5:

Theorem 6 For (a, z) ∈ D and m > m5, the dominant component of Theorems 3 and 4, namely Z π/2 √ 1 −a −z/2 −iω 2 mz cos ω c0 := r e Hm(a, z, e )e dω, 2π −π/2 can be given an eﬀective form for any order N ≥ 0, as N−1 1 X 1 Z π/2 √ c = r−ae−z/2 e−iωaA e2 mz cos ω dω 0 2π mh/2 h h=0 −π/2

+ Sn(a, z) E2,N , where the error term is bounded as ! N/2 2 2 θ π m5 π =(a) sec 2 1/2 θ E2,N = Θ : √ exp sec 2 4m 32pm|z| 2 and the Ah are to be calculated as the ﬁrst N coeﬃcients of

∞   X X ak  A xh := exp xk , h k h=0 k≥1  via (25) with v := e−iω.

25 Proof. Inserting the eﬀective H-kernel expansion from Theorem 5 directly into the c0 integral gives the indicated sum over h ∈ [0,N − 1] plus an error term

1 m N/2 Z π/2 √ Θ: r−ae−z/22 5 eω=(a)e−2 m|z| cos(θ/2) cos ω dω. 2π 4m −π/2

Now using Lemma 1 on cos ω and the V0-part of Lemma 2 we obtain the E2,N bound of the theorem. QED

With Theorem 6 we have come far enough to see that a Laguerre evaluation can be obtained—up to a sub-exponentially small relative error—via the Ah terms in said theorem. To this end, inspection of the defining relations reveals that in general we can −iω decompose an Ah coefficient in terms of powers of v := e , namely we define αh,µ terms via

h X h+2u Ah =: αh,u(a, z)v . (29) u=0 For example, α00 = 1, 1 + a α = − z1/2, 10 2 z3/2 α = , 11 12 1 α = 5a2 + 15a + 16 z5/2. 31 480

The point being, we now have special formulae for the dominant contribution c0, namely

N−1 h 1 X 1 X √ c = r−ae−z/2 α (a, z) I(2 mz, a + h + 2u) (30) 0 2π mh/2 h,u h=0 u=0 + Sn(a, z) E2,N , (31) where the integral

Z π/2 I(p, q) := e−iqωep cos ω dω (32) −π/2 thus emerges as a fundamental entity for the research at hand.

26 5 I-integrals and an “exp-arc” method

Having reduced the problem of sub-exponential Laguerre growth to a study of the I- integrals (32), we next develop a method that is effective for both their numerical and theoretical estimation. This method amounts to the avoidance of stationary-phase techniques, employing instead various forms of exponential-arcsin series, as we see shortly. Let us first define with a more general integral. For any complex pair (p, q), assume α, β ∈ (−π, π) and define Z β I(p, q, α, β) := e−iqωep cos ω dω, (33) α so that our special case (32) is simply

I(p, q) := I(p, q, −π/2, π/2).

An aside is relevant here. The I integral can be written in terms of an Anger function J and Weber function E—see [1]—as

iqπ/2 I(p, q) = πe (J−q(−ip) + iE−q(−ip)), with an important special case

I(p, 0) = π(J0(−ip) + L0(p)), where J here is the standard Bessel function and L denotes the modiﬁed Struve function [1]. Moreover, one may write the Bessel functions Jn of integer order n in terms of I-integrals: 1 J (z) = e−iπn/2I(iz, n) + eiπn/2I(−iz, n) , (34) n 2π and the modiﬁed Bessel function, again of integer order n: 1 I (z) = (I(z, n) + (−1)nI(−z, n)) , (35) n 2π about which representations we shall have more to say in a later section. These forms (34), (35) are easily derived directly from standard integral representations; see [1, 9.1.21], but note as with [1, 9.1.22], that non-integer n is more complicated because of a cut-term not unlike such terms we have encountered in the Laguerre problem for non-integer a parameter. (See relations (68, 69) for general indices ν on Jν,Iν.)

5.1 The exp-arc method explained Now we investigate what we call exponential-arcsine (“exp-arc”) series. First, for any complex τ and x ∈ [−1, 1], one has a remarkable expansion (see [4]):

∞ X xk eτ arcsin x = 1 + r (τ) , (36) k k! k=1

27 where the coeﬃcients depend on the parity of the index, as

m m Y 2 2 Y 2 2 r2m+1(τ) := τ τ + (2j − 1) , r2m(τ) := τ + (2j − 2) . j=1 j=1

By diﬀerentiating with respect to x we obtain

∞ eτ arcsin x 1 X xk √ = r (τ) , 2 τ k+1 k! 1 − x k=0 valid for x ∈ (−1, 1). In particular, we have the important expansion (here we deﬁne a function G, in passing)

∞ cosh(τ arcsin x) X x2k G(τ, x) := √ = g (τ) , (37) 2 k (2k)! 1 − x k=0 where k Y 2 2 gk(τ) := (2j − 1) + τ . j=1

The g coeﬃcients are especially easy to remember, as the sequence (g0, g1,... ) is simply (1, (12 + τ 2), (12 + τ 2)(32 + τ 2), (12 + τ 2)(32 + τ 2)(52 + τ 2),... ). These exp-arc expansions may be applied to the I(p, q, α, β) integrals as follows. From (33) and its subsequent manipulations we have

β Z 2 I(p, q, α, β) = ep e−iqωe−2p sin (ω/2) dω α

sin β −2iq arcsin x Z 2 e 2 = 2ep √ e−2px dx α 1 − x2 − sin 2

∞ Z sin β X rk+1(−2iq) 2 2 I(p, q, α, β) = iep x2ke−2px dx. (38) k! q − sin α k=0 2 It is sometimes quite useful to deﬁne also U(p, q, ψ) := I(p, q, 0, 2ψ), with the interesting partitioning U = U1 + U2 and respective series developments as follows:

∞ 2n n 1 X Y 4q2 U (p, q, ψ) = ep cos(2ψ) sin(2qψ) + 4p ep n 1 − β (p, ψ), 1 2q (2n + 1)4n (2k − 1)2 2n+1 n=0 k=1 ∞ n n−1 2 1 p cos(2ψ) 2 p X 4 Y q U2(p, q, ψ) := e sin (qψ) + 2pq e 1 − β2n(p, ψ), q n22n k2 n=1 n k=1

28 where β is an error-function-class integral Z sin ψ n+1 −2p x2 βn(p, ψ) := x e dx (39) 0 1 = (2p)−1−n/2 Γ(n/2 + 1) − Γ(n/2 + 1, 2p sin2 ψ) . 2 With these prescriptions we can always write I(p, q, α, β) = U(p, q, β/2) − U(p, q, α/2). (40) Even though we shall eventually be contemplating asymptotic expansions, convergence of such I-series is the rule: Lemma 11 For α, β ∈ (−π, π) and any complex pair (p, q), the series (38) converges absolutely.

Proof. By relation (40) is enough to show that for |ψ| < π/2 each of U1,U2 converges 2n −n √ n+2 2|<(p)| absolutely. Since n 4 = O(1/ n), while |βn(p, ψ| ≤ | sin ψ| e , and the two product terms are bounded, the absolute convergence of both U1,U2 is assured. QED Likewise, from (37), (38) we now have an absolutely convergent expansion for the special-case I integral: √ 1/ 2 Z 2 I(p, q) = 4ep G(−2iq, x)e−2px (41) 0 ∞ X gk(−2iq) = 4ep B (p), (42) (2k)! k k=0 where Bk is an error-function-class integral √ Z 1/ 2 2k −2px2 Bk(p) := β2k−1(p, π/4) = x e dx. (43) 0 It is both computationally and theoretically important that Bk can be given a closed form (in terms of Γ- and incomplete Γ-functions) as well as a recursion relation. Namely, we have

1 1 B (p) = {Γ(k + 1/2) − Γ(k + 1/2, p)} , (44) k 2 (2p)k+1/2 so that 1 √ r π √ B (p) = √ π − Γ(1/2, p) = erf ( p) . 0 8p 8p A recursion follows for k > 0: 2k − 1 2−k−3/2 B (p) := B − e−p, (45) k 4p k−1 p We shall have more to say later about the computational eﬃcacy of these relations. For the moment, we next focus upon theoretical applications relevant to the asymptotic nature of I-integrals.

29 5.2 Eﬀective expansion for the cosh-arc G-series Our asymptotic analysis of representation (41) begins with a lemma that reveals how the G function (37) has an attractive self-similarity property. Namely, the function appears naturally in modiﬁed form within its own error terms.

Lemma 12 For any complex τ the G function (37) can be given an eﬀective expansion to any integer order N ≥ 0, as cosh(τ arcsin x) G(τ, x) := √ (46) 1 − x2 N−1 X x2k x2N = g (τ) + g (τ) (1 + T (τ, x)) , (47) k (2k)! N (2N)! N k=0 √ with the error term TN conditionally bounded over real x ∈ [0, 1/ 2] in the form

T = Θ : 1 , if N ≥ 2x2|τ|2 − 1; N √ = Θ : ( 2 x2 + x|τ|)e|τ| arcsin x , otherwise.

Proof. The error term is, by the deﬁnition of the gk(τ), given by the absolutely convergent sum TN = hN + hN hN+1 + hN hN+1hN+2 + ..., where (2k + 1)2 + τ 2 h := x2. k (2k + 1)(2k + 2) When |τ|2x2 ≤ (N + 1)/2, and since x2 ≤ 1/2, it is immediate that for k ≥ N we have a bound: (4k2 + 4k + 1)/2 + (N + 1)/2 |h | ≤ ≤ 1/2, k 4k2 + 5k + 2 whence TN = Θ : 1/2+1/4+1/8+... , settling the ﬁrst√ conditional bound of the theorem. In any case—i.e. any complex τ and any x ∈ [0, 1/ 2], we have for j ≥ 0:

2 j (2N + 2k + 1)2 (2k + 1)2 + τ 2 (2k+1) Y (2N+2k+1)2 h h ··· h = x2j N N+1 N+j (2N + 2k + 1)(2N + 2k + 2)(2k + 1)2 k=0 g (|τ|) = Θ : j+1 . (2j + 1)!!2 However, it is elementary that (2j + 1)!!2 ≥ (2j + 1)! by simple factor-tallying, so

12 + |τ|2 · 2 12 + |τ|2 32 + |τ|2 · 4 |T | ≤ x2 + x4 + ... N 2! 4! ∂ = x G(|τ|, x), ∂x

30 where we have noticed that the right-hand series here is itself a diﬀerentiated “cosh-arc” series. Thus

2 ∂ cosh(|τ| arcsin x) x u −u x|τ| u −u TN = Θ : x √ = Θ : e + e + e − e ∂x 1 − x2 2 (1 − x2)3/2 2 (1 − x2)

2 2 where u := |τ| arcsin x. Now by excluding 2x |τ| ≤ N+1 for the second conditional bound√ of the lemma, we have |τ| ≥ 1, whence the e−u terms can be ignored over x ∈ [0, 1/ 2], and the second conditional bound follows. QED

5.3 Eﬀective expansion of the I-integral We need one more set of brief lemmas, all elementary but useful in regard to asymptotic analysis, including analysis that reaches beyond the present treatment. Consider the standard gamma function’s common representation Z ∞ Γ(a, z) := ta−1e−t dt, z for z > 0 and an alternative form, obtained by change of variables Z ∞ Γ(a, z) = zae−z e−zs(1 + s)a−1 ds, 0 the later integral representation being valid at least for conditions ((<(z) > 0) and a ∈ C) or ((<(z) = 0) and (<(a) < 0)).

Lemma 13 If real ρ ≥ 0 and <(z) > 0,

Z ∞ e−zs 2 ρ ds = Θ . 0 (1 + s) |z| Accordingly, for real a < 1, Γ(a, z) = Θ 2|z|a−1e−z .

Proof. For ρ = 0 the result is trivial. For ρ > 0, integration by parts gives the integral as 1/z + (ρ/z)Θ(1/ρ). The second result follows immediately from the alternative representation above. QED

The next lemma uses a standard gamma-recursion to bring gamma arguments of interest into the zone of applicability of Lemma 13:

31 Lemma 14 For real a, integer m ≥ max(0, a − 1), <(z) ≥ 0, z 6= 0, we have an eﬀective expansion

(a − 1) (a − 1) ··· (a − m + 1) (a − 1) ··· (a − m) Γ(a, z) = za−1e−z 1 + + ··· + + Θ : 2 · . z zm−1 |z|m

In particular, for integer M ≥ 0, <(z) ≥ 0, z 6= 0 and |z| ≥ 2M − 1, we have Γ(M + 1/2, z) = Θ 2|z|M−1/2e−z .

Remark: Results such as these on incomplete-gamma errors appear in various texts on special functions, e.g., [22, §. 2.2, p. 110]. In the present treatment we have stated the bounds in a manner and style consistent with the rest of the present analysis. Proof. The well known incomplete-gamma recursion, [1], for integer-depth m ≥ 0 yields

(a − 1) (a − 1) ··· (a − m + 1) Γ(a, z) = za−1e−z 1 + + ··· + z zm−1

+(a − 1) ··· (a − m)Γ(a − m, z). For real a < 1, take m ≥ 0, whence Lemma 13 gives the error term as Θ : 2(a − 1) ··· (a − m)za−m−1e−z. Otherwise, take m = ba + 1c and |z| > 2(a − 1) in which case

Γ(a, z) = za−1e−z · Θ : 1 + 1/2 + 1/22 + ··· + 1/2m−1 + 2/2m and the sum here is Θ(2). Finally, the assignment a := M + 1/2 yields the ﬁnal theta bound as corollary. QED

The results of the present section may now be applied to a general, eﬀective expansion of the I-integral whenever <(p) is suﬃciently positive.

Theorem 7 (Eﬀective I expansion) For the integral

Z π/2 I(p, q) := I(p, q, −π/2, π/2) = e−iqωep cos ω dω, −π/2 assume an integer expansion order N ≥ 1. Assume φ := arg(p) ∈ (−π/2, π/2) and conditions <(p) ≥ 2N + 1, <(p) ≥ 2π|q|2. Then we have an eﬀective expansion

r ( N−1 r 2π X gk(−2iq) 1 8 π I(p, q) = ep + Θ : e−p cosh |q| (48) p k! 8k pk πp 2 k=0 g (−2iq) 1 + N 1 + Θ : u secN+1/2 φ , (49) N! 8N pN N

32 where the gk are as in the cosh-arc expansion (37), and we may take 2N+1Γ(N + 1) u := 1 + 2N + √ ; (50) N πΓ(N + 1/2)

2 however, with the extra condition N ≥ 4|q| − 1, taking uN := 1 suﬃces.

Proof. Insertion of the series of Lemma 12 into representation (41) results in √ ( N Z 1/ 2 ) X gk(−2iq) gN (−2iq) 2 I(p, q) = 4ep B (p) + x2N T (−2iq, x)e−2px dx , (2k)! k (2N)! N k=0 0 where the Bk(p) are given by (43). The sum over k ∈ [0,N] here is thus

N 1 X gk(−2iq) Γ(k + 1/2) 2 (2k)! (2p)k+1/2 k=0

N X |gk(−2iq)| 1 + Θ : e−p , (2k)! 2k+1/2p k=0 where the Θ-term here follows from Lemma 14 on our condition <(p) ≥ 2N + 1. But this very Θ-term is bounded above by

−p N e X gk(2|q|) 1 √ p 2 (2k)! 2k k=0 e−p 1 = cosh 2|q| arcsin √ , p 2 so we have settled the summation and the cosh(π|q|/2) term in (48). Now consider the integral term √ Z 1/ 2 2N −2px2 I0 := x TN (−2iq, x)e dx. 0 √ −1p Deﬁne γ := |q| (N + 1)/8. If γ ≥ 1/ 2 then by Lemma 12√ we know TN = Θ(1) and our theorem follows in the uN := 1 case. Otherwise, γ < 1/ 2 and we bound I0 using two integrals √ √ Z 1/ 2 Z 1/ 2 √ 2N −2<(p)x2 2N+2 2N+1 2|q| arcsin x−2<(p)x2 |I0| ≤ x e dx + 2 x + 2|q| x e dx. 0 γ √ 2 From Lemma 1 we know that the√ exponent here can be taken to be π|q|x/ 2 − 2<(p)x . For the assignments α := π|q|/ 8, β := 2<(p) we have β > 4α/γ so that by Lemma 2 we have a bound 1 Γ(N + 1/2) √ I ≤ + 2 V (α, β, γ) + 2|q| V (α, β, γ) 0 2 (2<(p))N+1/2 N+1 N+1/2

33 1 Γ(N + 1/2) Γ(N + 3/2) Γ(N + 1) ≤ + 2N + 2N+1|q| . 2 (2<(p))N+1/2 (2<(p))N+3/2 (2<(p))N+1 Using |q|2 ≤ <(p)/(2π), <(p) ≥ 2N + 1, and writing <(p) = p cos φ yields the theorem with the N-dependent uN form. QED

Note that the second (cosh) term of the expansion in Theorem 7 is relatively exponentially small, in that it decays like e−<(p), while the last term is a typical, eﬀective “correction” to the asymptotic sum over k ∈ [0,N].

(−a) 6 Eﬀective asymptotics for Ln (−z)

At last we are in a position to provide explicit terms for Laguerre expansions in the sub- exponential-growth regime, which regime turns out to be precisely characterized by the parameter-pair requirement: (a, z) ∈ D. First, for convenience we recapitulate the thresholds for suﬃciently large m := n + 1 from our previous theorems, and add some more:

2 m0 := |z|/4, m1 := 5 |<(z)| + (|=(a)| + |=(z)|/2) ,

2 m2 := <(a), m3 := −(5/4)|z| cos θ, m4 = 4|z|, m5 := |z|(1 + |a| + |z|/2) , 2 2 2 4 m6 := (2N + 1) sec (θ/2)/|z|, m7 := 4π sec (θ/2)(|q| + 3N − 3) .

Here, θ := arg(z) as before, while m6, m7 assume an asymptotic expansion order N ≥ 1 in what follows. (We are aware that some of the mi are masked by others; however it is best to assume all of the mi are in force, because previous, partial results do depend on particular thresholds.) Before delving into our main result, let us remind ourselves of previous nomenclature:

k Y 2 2 gk(τ) := (2j − 1) + τ , (51) j=1 √ √ v z k k v z k+2 a := (1 + a)(−1)k + 1 − (−1)k , (52) k 2 k + 2 2 ∞   X X ak  A xh := exp xk , (53) h k h=0 k≥1  h X h+2u Ah =: αh,u(a, z)v . (54) u=0

Note that the αh,u coefficients are thus implicitly defined, in terms of the original ak h+2u functions. Notationally, we may write αh,u = [v ]Ah(v), to indicate that we select the coefficient of vh+2u.

34 6.1 The general sub-exponential expansion √ Using Theorem 7 with p := 2 mz, and inserting this into formula (30), we arrive at our desired eﬀective Laguerre expansion.

Theorem 8 (Eﬀective Laguerre expansion) Assume (a, z) ∈ D. For asymptotic expansion order N ≥ 1, and m := n + 1 suﬃciently large in the sense m > maxi∈[0,7] mi, we have the expansion

√   −z/2 2 mz N−1 1 e e X Cj CN  L(−a)(−z) = √ + + E + E , (55) n 2 π z1/4−a/2 m1/4+a/2 mj/2 mN/2 1 3,N  j=0  where the expansion coeﬃcients Cj are given ﬁnitely by

j j−k X 1 1 X C := α (a, z) g (−2i(a + j − k + 2u)), (56) j 16k k! zk/2 j−k,u k k=0 u=0 in terms of (51) and (53), while the error term CN is bounded as

N N−v X 1 θ X C = Θ : 1 + u secv+1/2 |α (a, z) g (−2i(a + N − v + 2u))| N |z|v/216vv! v 2 N−v,u v v=1 u=0 θ + Θ : 4(m /4)N/2 sec1/2 , 5 2 with uv taking the v-dependent form of (50) in Theorem 7. The term E1 is sub-exponentially small (from Theorem 4), as is

s N−1 h 4 √ X 1 X π E = Θ : √ e−2 mz |α (a, z)| cosh |a + h + 2u| . 3,N π mz mh/2 h,u 2 h=0 u=0

Proof. For brevity we leave out the details—all of which are straightforward, if tedious applications of the previous theorems and formulae. QED We now have a resolution of the domain of sub-exponential growth, as:

Corollary 1 For (a, z) ∈ D, the Laguerre polynomial grows sub-exponentially, in the sense that for order N ≥ 1, and any ε > 0,   N−1 √ X Cj 1  L(−a)(−z) = S (a, z) + O + O e−(2−ε) m|z| cos(θ/2) , n n mj/2 mN/2  j=0  with all coeﬃcients and the implied big-O constant eﬀectively bounded via our previous theorems. Moreover, for (a, z) 6∈ D, the large-n growth is not sub-exponential. Thus, the precise domain of sub-exponential growth is characterized by (a, z) ∈ D.

35 Proof. The given sub-exponential formula is a paraphrase of Theorem 8. Now assume z ∈ (−∞, 0]. We already know z = 0 does not yield such growth (see (2). Now for z negative real, note that the integrals in (17, 19) are both decaying in large m. Finally, √ the integral in (16) has phase factor | exp(2 mz cos ω)| ≤ 1, and Lemma 6 show that c1 also cannot grow sub-exponentially in m. QED

This corollary echoes, of course, the classical Perron result (7), and we again admit that historical efforts derived the Cj coefficients in principle. What we have done up to this point is a) established such Laguerre asymptotics over the full sub-exponential-growth domain of (a, z) ∈ D; b) established rigorous, explicit (i.e. effective) errors over said domain;

c) provided in passing an algorithm for generating the Cj and, perforce, eﬀective bounding constants. We now turn to the algorithmic and computational aspects of the general asymptotics.

6.2 Algorithm for explicit asymptotic coeﬃcients

Theorem 8 indicates that, to obtain actual Cj coeﬃcients, we need the cosh-arc numbers gk(τ) and the αh,u(a, z) coeﬃcients. Observe that the chain of implications (51) amounts to an algorithm for generation of the Ck. All of this can proceed via symbolic processing, noting that v is simply a place-holder throughout. Remarkably, there is another approach—a fast algorithm that bypasses much of the symbolic tedium. First, we have an explicit recursion for Ah, with A0 := 1, as

k−1 1 X A = A a (57) k k j k−j j=0 as follows from diﬀerentiating (53) logarithmically, and then comparing terms. Second, when we use (54) together with (57), we obtain a recursion devoid of the symbolic place- holder v, as

k−1 1 X α = (α b + α d ) , (58) k,u k j,u k−j j,u−1 k−j j=0 where these new recursion coeﬃcients are √ z h b := (−1)h(1 + a) , h 2 √ h z h+2 d := 1 − (−1)h . h h + 2 2

36 It is important that in practice we deﬁne α0,0 := 1 and force any αj,u with u > j or u < 0 to vanish. In this sense, the collection of αk,u make up a lower-triangular matrix, e.g. the entries for k ≤ 3 appear thus:

 1 0 0 0  (a+1) 1/2 1 3/2  − 2 z 12 z 0 0   2  ,  a +3a+2 (a+1) 2 1 3   8 z − 24 z 288 z 0  a3+6a2+11a+6 3/2 5a2+15a+16 5/2 a+1 7/2 1 9/2 − 48 z 480 z − 576 z 10368 z

where α3,3 is the lower-right element here. These observations lead to a fast algorithm for computation of the asymptotic coeﬃ- cients:

− − − − − Algorithm (Fast computation of Laguerre coeﬃcients.) For given (a, z) ∈ D and desired expansion order N, this algorithm returns the asymptotic coeﬃcients (Ck : k ∈ [0,N]) of relation (56), Theorem 8.

1) Set α0,0 := 1 and for desired order N, calculate the lower-trianglar matrix elements (αk,u : 0 ≤ u ≤ k ≤ N) via a recursion such as α(k, u){ if(k == 0) return δ0,u; 1 Pk−1 return k j=0 (αj,ubk−j + αj,u−1dk−j); } or use an unrolled, equivalent loop (i.e., one may generate the left-hand column of the α-matrix, then ﬁll in one row at a time, lexicographically). 2 2 2) Use the recursion gk(τ) = (2k + 1) + τ gk−1(τ) and the lower-triangular matrix of α values to generate the C coeﬃcients via (56).

− − − − −

We observe that the expensive sums in this algorithm are all acyclic convolutions. Thus, for numerical input (a, z) the algorithm complexity turns out to be O N 2+ arithmetic operations, with the “2” part of the complexity power arising from the area of the lower- triangular sector.7 We employed the algorithm to generate exact asymptotic coeﬃcients as follows:

C0 = 1, −12a2 − 24za + 4z2 − 24z + 3 C = √ , 1 48 z

7For example, ﬂoating-point FFT-based convolutions of length L require O(L log L) operations, and we are calling that O L1+.

37 1 C = 144a4 + 576za3 + 480z2a2 + 1728za2 − 360a2 − 192z3a 2 4608z + 1152z2a + 1584za + 16z4 − 192z3 + 312z2 + 432z + 81 , 1 C = −8640a6 − 51840za5 − 95040z2a4 − 259200za4 + 75600a4 3 3317760z3/2 −34560z3a3 − 622080z2a3 − 388800za3 + 31680z4a2 − 103680z3a2 − 1395360z2a2 −129600za2 − 139860a2 − 5760z5a + 69120z4a + 60480z3a − 1192320z2a + 100440za +320z6 − 5760z5 + 7632z4 + 129600z3 − 267300z2 + 48600z + 30375 , and so on. Note that the C1 form here agrees with the PAMO coeﬃcient in (9). We were able to generate the full, symbolic C64(a, z) in about one minute of CPU on a typical desktop computer. To aid future researchers, we report that numerator of C64 has degree 128 in both a, z, while the denominator is

519667715662517012461660216412794662396986519336129512687040904512259

134698859029589268254378668537757499729050302014891552685362781283752 17277745741145251371621775626061919271284199391232000000000000000000 0000000 It is not unexpected that every prime number not exceeding 64 divides this denominator. Attempts to simplify the formidable, bivariate Ck(a, z) coeﬃcients led us also to the following:

Conjecture 1 Given index k ≥ 1, consider the speciﬁc parameter assignments a in {−1/2, −3/2,..., −1/2 − (k − 1)}. Then for any such value of a, the coeﬃcient Ck(a, z) factors into a positive power of z times a rational polynomial in z, such that the total degree in z is 3k/2.

For instance with k = 4 we obtain, with total degree 6 in each case:

z2 5z4 + 300z3 + 4842z2 + 14580z − 63747 C (−7/2, z) = . 4 2488320 z2 5z4 + 180z3 + 882z2 − 10692z − 39771 C (−5/2, z) = , 4 2488320 z 5z5 + 60z4 − 1038z3 − 6804z2 + 17253z + 29160 C (−3/2, z) = , 4 2488320 z 5z5 − 60z4 − 918z3 + 6804z2 + 16605z − 48600 C (−1/2, z) = . 4 2488320 It seems very likely that Conjecture 1 is a harbinger of much more subtle structure that we have not yet been able to abstract.

38 6.3 Generating and verifying eﬀective bounds

For the first nontrivial effective bound, we can employ the rigorous bound C1 in Theorem 8, with N = 1, to get an effective version of the original asymptotic (3), as

C L(−a)(−z) = S (a, z) 1 + √ 1 + E + E , n n m 1 3,1 with 1 − 4a2 θ θ C = Θ : 1 + 6 sec3/2 + 2|z|1/2(1 + |a| + |z|/2) sec1/2 , 1 16|z|1/2 2 2 and we remind ourselves that E , E are both sub-exponentially small. 1 3,1 √ At last we have an effective numerator, then, for the 1/ m asymptotic term. Though this effective numerator is almost surely nonoptimal, we are evidently on the right track, because the exact C1 asymptotic coefficient above has very much the same form as does our C1 here (i.e., same degrees of appearance for a, z, and similar coefficients). And, it is easy to see that C1 is an upper bound on |C1| itself, as must of course be true. In spite of the unwieldy character of the exact Ck coefficients, it is possible to verify numerically the asymptotic expansion, at least to a few orders. One good worked example is to take order N = 4, giving the partial series

L(−a)(−z) C C C n ∼ 1 + 1 + 2 + 3 1/2 3/2 Sn(a, z) m m m and compare direct summation of (1) with the right-hand asymptotic piece. The results 64 are (note that the Sn denominator here is of order 10 ):

L(−1−i)(−1 + i) 5000 ≈ 0.98514574 − 0.0080754i, S5000 from the direct sum, while the righthand-side sum over the m−k/2 gives

≈ 0.98514577 − 0.0080755i.

This amounts to an absolute error of ≈ 9.80·10−8, or more usefully stated, ≈ 2.45187/m2. The experimental situation is good, since the first missing asymptotic coefficient is C4(1 + i, 1 − i) ≈ 2.48. However, regarding rigor, more important than these good approximations is the effective coefficient C4, which for the current parameters we get from Theorem 8 as

C4(1 + i, 1 − i) = Θ : 202.63. This amounts to a little over 2 decimals of penalty, in trade for rigor. If we ignore the sub-exponentially small error terms E1, E3,4 (which are well below the 7-digit accuracy threshold) then we ﬁnd that our 7-decimal-accurate asymptotic piece is rigorously correct to 4 places. In a word: One can prove in this way that

(−1−i) 64 L5000 (−1 + i) = ((0.8053 ± 0.0001) + (1.1483 ± 0.0001) i) · 10 ,

39 and it should be no surprise that this partial-but-rigorous asymptotic approach is radically faster—for such large n—than naive direct summation.8 So this is our technique for obtaining provable results about the large-n behavior of Ln. We have mentioned research motives in Section 1.1. One additional application might be to use such effective bounds to rule out zeros of L in the crossed (a, z)-plane, that is, n √ to locate sufficiently large m such that the first terms 1 + C1/ m cannot be overwhelmed by the rest of the terms.

7 The exp-arc method and oscillatory behavior

Though the exp-arc method has succeeded in establishing rigorous asymptotics for sub- exponential growth, there remains the issue of the Fej´erform (6) for z 6= 0 on the cut (−∞, 0]. Such oscillatory behavior can indeed be dealt with, but separate techniques come into play. For one thing, contour integrals must be handled diﬀerently.

7.1 Remarks on the oscillatory Laguerre regime For this next analysis we shall proceed non-rigorously, in that eﬀective bounds are problematic for z negative real. Yet, we shall still gain insight, and ﬁnd some ways to provide at least some low-order bounds. We remind ourselves that even on z ∈ (−∞, 0) the contour prescription of Figure 1 is p valid, and the Laguerre polynomial is exactly the sum c1+d1+e1, with R := m/|z| > 1/2 being the only requirement for contour validity. However—and this is important—the dominant contribution (22) has to change, to involve an expanded integration interval; in fact, now we must use the contour integral c1 itself as the main contribution:

Z 3π/2 √ 1 −a −z/2 −iω 2 mz cos ω c1 := r e Hm a, z, e e dω, (59) 2π −π/2

where we have used our convention θ := arg(z) = π for negative real z. This wider integration range is necessary because the procedure of Theorem 3, which peeled oﬀ the √ sub-exponentially small c2 integral, fails when exp(2 mz cos ω) has constant magnitude, as it does for negative real z. Another way to view this is that we need to include two stationary points for cos ω in the integration interval, namely ω = 0, π. These observations lead to an analysis of the wider integral

Z 3π/2 e−iqωep cos ω dω = I(p, q) + e−iπq I(−p, q), −π/2

easily established by bisecting the range [−π/2, 3π/2]. In turn, the main contributor to

8We recall that the direct series (1) admits of various accelerations.

40 Laguerre evaluations should be, rather than (30), the form

N−1 h 1 X 1 X c = S (a, z) E + r−ae−z/2 α (a, z) (60) 1 n 4,N 2π mh/2 h,u h=0 u=0 n √ √ o × I(2 mz, a + h + 2u) + e−iπ(a+h+2u)I(−2 mz, a + h + 2u)

where E4,N is an N-th order error term which we shall not calculate here. These machinations do correctly give the leading (cosine) term of the classical Fejér expansion (6), and presumably, in analogy with Theorem 8, yield closed forms for the general coefficients in the Perron form for negative real z [30, Theorem 8.22.2]. Because we are not claiming rigor in this subsection—and especially as we do not yet have a comprehensive exp-arc approach to effective bounds in this case—we simply claim without proof, on the basis of (60), that the correct generalization of the Fejér series (6) for z on the open negative cut (−∞, 0), and any complex a, is

e−z/2 L(−a)(−z) ∼ √ (61) n π(−z)1/4−a/2 m1/4+a/2 ( ∞ ! ∞ ! ) X Ak √ X Bk √ × cos 2 −mz + aπ/2 − π/4 + sin 2 −mz + aπ/2 − π/4 , mk mk+1/2 k=0 k=0 where these oscillatory-series coeﬃcients are directly related to the coeﬃcients in Theorem 8 by Ak := C2k(a, z),

Bk := iC2k+1(a, z).

It turns out that for a real, every Ak,Bk is then real, whence the asymptotic has all real terms. An important observation is relevant here: The aforementioned SzegöTheorem 8.22.2 for the oscillatory Laguerre mode is stated in a fashion structurally different from our asymptotic (61); for one thing we are conjecturing that Sezgö’sown Aodd,Beven— which are not defined quite like ours here—vanish.9 We did perform numerical experiments on the expansion (61). A worked example is for n = 2880, a = −1 + i, z = −16, for which we calculated the prefactor P , the obvious angle χ, and the coefficients A0,A1,A2,B0,B1 to obtain C C iC iC L(1−i)(16) ≈ P · 1 + 2 + 4 cos χ + 1 + 3 sin χ 2880 m m2 m1/2 m3/2 = (2.30 ... ) + (0.67839 ... ) i, correct to the implied precision. Note that the imaginary part is considerably more accurate—one of the typical complicating effects of allowing complex a parameter—and signaling the considerable difficulty of effective oscillatory expansions for general parameters. 9Our oscillatory asymptotic (61) has been verified to several terms by N. Temme; also he verifies our claim that the classical Szegöcoefficients do vanish for the parities indicated.

41 7.2 Application of the exp-arc method to Bessel functions The history of Bessel-function asymptotics is one of the great success stories in the annals of analysis. As early as 1823, Poisson developed the beginnings of Bessel asymptotics [36] [32], and eventually Hankel developed the classic, complete asymptotic series [1, §9.2.5], and eﬀective bounds on error terms (for certain parameter domains) are well known, and in many cases optimal [36]. It is instructive to explore, at least partially, the application of our exp-arc method to Bessel expansions. We recall (34), (35) which we repeat here: 1 J (z) = e−iπn/2I(iz, n) + eiπn/2I(−iz, n) , (62) n 2π

1 I (z) = (I(z, n) + (−1)nI(−z, n)) , (63) n 2π both valid for integer n.

7.3 Asymptotics for Bessel functions In,Jn Because our present methods are geared toward large-n growth, the modiﬁed Bessel function In, which for real argument is non-oscillatory, is easier to analyze with the exp-arc method. Theorem 7 implies an asymptotic expansion, based on our absolutely convergent series for Re(z) > 0, according to

2 X gk(−2in) I (z) = ezB (z) + (−1)ne−zB (−z) (64) n π (2k)! k k k≥0 z ∞ e X gk(−2in) 1 ∼ √ , 2πz k! 8k zk k=0 where the ﬁrst sum is convergent, exact, and the second sum agrees with the classical Hankel asymptotic [36] [1]. Moreover, under the conditions θ := arg(z) ∈ (−π/2, π/2), <(z) ≥ max 2N + 1, 2πn2, and N > 4n2 − 1, the error on truncating the asymptotic series at N − 1 summands inclusive is, again by Theorem 7, g (−2in) 1 Θ: N 1 + secN+1/2 θ , N! 8N zN plus, of course, some sub-exponential terms. Thus, typical rigorous error bounds in the literature are recovered, or at least suggested (e.g., real positive z yields a classic result that the ﬁrst missing term with z−N carries only a factor-of-2 magnitude penalty). Now to a brief analysis of frank oscillatory behavior. Asymptotics for Jn(z), <(z) > 0 are possible via the exp-arc approach, but the details are rather intricate. Let us give at least an example of how an exp-arc series might be used to establish bounds. First, there is a useful lemma that often applies in oscillatory cases:

42 Lemma 15 Let F (x) be real, twice diﬀerentiable, with |F 00(x)| ≥ ρ > 0 for x on real interval (a, b). If a real function G(x) on said interval has G/F 0 is monotonic, |G| ≤ M, then Z b 8M G(x)eiF (x) dx = Θ : √ . a ρ

Proof. This is proved in [33]. QED

Then, we follow with an implication speciﬁc to the present work:

Lemma 16 For any complex p with <(p) ≥ 0, and positive real ν, we have √ Z 1/ 2 2ν −2px2 9 1 Bν(p) := x e dx = Θ : ν p . 0 2 |p|

Proof. We establish two diﬀerent upper bounds. First, if <(p) > 0 the integral is bounded as Z ∞ r 1 −2<(p)x2 1 π Bν(p) = Θ : ν e dx = ν+1 . 2 0 2 2<(p) Second, if =(p) > 0 we may use Lemma 15, with G := x2νe−2<(p)x2 and F := −2=(p)x2. Now this G has at most two branches of monotonicity, so we conclude 1 8 Bν(p) = Θ : . 2ν p|=(p)|

Then we simply observe that 9 · 2−ν|p|−1/2 is always an upper bound for one of these two bounds. QED

With these lemmas in hand, we can peel oﬀ some desired number of summands from (41) and perform integration by parts on the tail of the sum, as in the following example where just one term is peeled oﬀ: √ Z 1/ 2 ! p −2px2 I(p, q) = 4e B0(p) + f(x)e dx , (65) 0 cos(2q arcsin x) f(x) := √ − 1. 1 − x2 Then integration by parts yields √ √ 1/ 2 Z 1/ 2 ! p 1 f(x) 1 d f(x) I(p, q) = 4e B0(p) − + dx . 4p x x=0 4p 0 dx x

These machinations lead to

43 Lemma 17 For <(p) > 0 and any complex q we have a ﬁrst-order expansion with eﬀective error bound: √ r2π 2 π 2 ep I(p, q) = ep − cos q + Θ : + Θ : C(q) , p p 2 p2 p3/2 where X |gk(−2iq)| (2k − 1) C(q) := 9 . (2k)! 2k k≥1

Proof. The proof is a straightforward application of relation (44), Lemma 14, and Lemma (16). QED With the above ideas we have, at least, a ﬁrst-order eﬀective bound for Bessel functions in the oscillatory regime:

Theorem 9 For integer n and <(z) > 0, we have an eﬀective Bessel asymptotic, in the form r 2 π π J (z) = cos z − n − n πz 2 4 √ 2 1 +Θ : π |z|2 D(n) +Θ : , |z|3/2 where 9 X |gk(−2in)| (2k − 1) D(n) := . π (2k)! 2k k≥1

Moreover, D(n) can be given a closed√ form. In particular, for J0(z) the k-sum evaluates to 2, while for J1(z) the sum is 4 2 − 2.

Proof. The proof follows from the sum (62) and Lemma 17. The specific evaluations of the k-sum arise from analyzing (d/dx)f/x in relation (65), noting that because of the structure of the gk, one only need find the k where gk changes sign, thereby evaluating at most two sums without absolute-valuing every summand. QED Is this Theorem 9 as strong as historical knoeledge, such as the known effective bounds on the celebrated Hankel expansion—as expounded by Watson [36, Ch. VII]? No, but we have shown that effective bounds in the presence of oscillation can indeed arise from the exp-arc approach. Theorem 9 is thus displayed to convey the basic ideas; extension of our 1st-order effective term to higher orders, via the exp-arc method per se, remains an open research problem.

44 7.4 Geometrically convergent algorithms for Jn Computationalists have know for decades that one way to evaluate Bessel functions uni- formly in the argument z is to use the standard ascending series for small |z|, but an asymptotic series for large |z|. However, via the exp-arc method one can establish a converging series whose evaluation only involves a single error-function evaluation, followed by recursion and elementary algebra. In fact, the relations (44), (45) can be used to calculate Jn(z) from relation (62) via the sum ∞ 2 X J (z) = g (−2in)(b cos χ − c sin χ) , (66) n π k k k k=0 with angle χ := z − πn/2 − π/4,

and the coeﬃcients bk, ck determined by iπ/4 −iπ/4 bk := Bk(iz)e + Bk(−iz)e , iπ/4 −iπ/4 ick := Bk(iz)e − Bk(−iz)e .

Note that if z is real then each bk, ck is real, whence our series here has all real terms. Note that our recursion (45) likewise ignites a recursion amongst the bk, ck. Note that (66) is actually the classical Hankel asymptotic if we replace Bk by its ﬁrst term in (44), namely (1/2)(2iz)−k−1/2Γ(k + 1/2); however, we already know that the sum (66) is always convergent. It is remakakble that we are using the same structure as the classical asymptotic, yet convergence for all complex z is guaranteed. Moreover, the Bk(iz) are independent of the order n and so can be re-used if multiple Jn(z) are desired for ﬁxed z. As just one experiment, we found that J12(8008 + 45i) is evaluated to 25 good decimals. using 60 terms (i.e. k ∈ [0, 60]) of the convergent sum. The primary points are a) the scheme is unconditionally convergent, and√ b) the bk, ck can be rapidly evaluated via recursion, after a single evaluation of erf iz. Incidentally, erf can be calculated via continued fraction, if one wants to avoid recourse to the standard ascending-asymptotic change of gears.10 In this way, one has a universal Bessel-computation scheme happily devoid of asymptotic accuracy and stability issues.

7.5 Hadamard expansions and the work of R. Paris Those acquainted with the intricacies of Bessel theory may observe that convergent expansion (64) is at least reminiscent of the convergent Hadamard expansion found in [36, p.204] for the modiﬁed Bessel function Iν, as z ∞ 1 e X 2 − ν k Iν(z) = √ {Γ(k + ν + 1/2) − Γ(k + ν + 1/2, 2z)} (67) 2πz Γ ν + 1 (2z)kk! 2 k=0 10One might argue that due to possible asymptotic schemes for incomplete-gamma (i.e. erf()), one is not truly avoiding the pitfalls of asymptotics. But this is wrong: It is known that incomplete-gamma (erf()) can be evaluated with always converging continued fractions, for example.

45 No. of summands K |In − (Hadamard)| |In − (exparc)| 32 10−7 10−13 64 10−9 10−23 128 10−10 10−42 256 10−11 10−81

Table 1: Comparison of Hadamard series (67) and exp-arc series (64) errors for Bessel evaluation −n−1/2 In(10); n = 4. The Hadamard series error behaves as K while the exp-arc error behaves as K−n−1/22−K (which is geometrical (linear) convergence) [23].

Though both series are absolutely convergent, here are some important diﬀerences between this Hadamard expansion and our exp-arc forms (64, 66). For example we have given our convergent sum only for integer ν. Moreover, the exp-arc expansion is geometrically convergent, while the Hadamard expansion is genuinely slower. Also note the geometrically convergent analogue (66) for Jν, integer ν. To be more concrete for the Bessel evaluation of I4(10), we have displayed various respective errors in Table 1. The whole research area of convergent expansions related to classical, asymptotic ones has been pioneered in large part by R. Paris, whose works cover real and complex domains, saddle points, and the like [24] [25] [26] [27]. We should point out that Paris was able to develop within the last decade some similar, linearly convergent Bessel series by modifying the “tails” of Hadamard-class series. One open research problem is: What transformations move between Paris’ convergent Bessel series and the exp-arc forms? One confounding aspect of this question is that the exp-arc method is only developed, so far, for integer indices. Also interesting: It may well be possible to develop eﬀective bounds—of the type in the present treatment—along the lines of the Paris theory. For general indices ν, it may also be possible to provide an exp-arc series using a representation valid for all cases (<(z) > 0) or (<(z) = 0 and <(ν) > 0), namely [36, p. 176]

Z π Z ∞ 1 sin(νπ) −νt−z sinh t Jν(z) = cos(νt − z sin t) dt − e dt, (68) π 0 π 0 with a corresponding representation Z π Z ∞ 1 z cos t sin(νπ) −νt−z cosh t Iν(z) = e cos(νt) dt − e dt, (69) π 0 π 0 itself valid for the same cases of z, ν. One wonders whether an exp-arc approach can be used to resolve the integrals here—which contribute when ν is not an integer—as exp-arc series.

46 8 Open problems

• How might one proceed with the exp-arc theory to obtain effective error bounds for oscillatory Laguerre modes, and-or oscillatory Bessel modes? We know that previous researchers have described how to give effective bounds in these cases (e.g., to our asymptotic (61), as in [30, §8.72]) but once again we stress: How can this be done explicitly, and for full parameter ranges? • Though we did provide an analytic/symbolic algorithm for such, is there any hope for a fully closed form for the asymptotic-series coefficients Ck(a, z) in Ln/Sn ∼ 1/2 3/2 1 + C1/m + C2/m + C3/m + ... ? And what about our Conjecture 1? (−a) • Where are the zeros in the complex z-plane—for fixed a—of Ln (−z)? Are “most of” the zeros along some a-dependent ray, in some sense? Note that effective error bounds conceivably could help in addressing this problem—by ruling out vast zero-free regions. There is a considerable literature on this zero-free-region topic, especially for polynomials in real variables. For example, with a := 0 the Laguerre zeros are all real and negative; see [20, Ch. X] and references therein. There is also an interesting connection between Laguerre zeros and eigenvalues of certain (large) matrices [10]. • How can the discrete iteration (10) be used directly to glean information about sub-exponential growth? One would think that insertion of a formal asymptotic form into the recursion would force certain relations between coefficients—but this is easier said than done, at least for the current authors. One may ask the same question for the Laguerre differential equation (12) as starting point. A promising research avenue for a discrete-iterative approach to asymptotics is [38].

• It would be useful to establish the very most efficient way to calculate Jn(z) with our converging series (66) and to know, for given arguments n, z how many terms of the exp-arc sum yield b good bits in the answer for Jn(z). It should also be possible to extract the classical ascending series for Jn directly from our converging series. • Can the integral pieces of (68, 69) be resolved as exp-arc series, to provide even more general, universally convergent I,J series (i.e. for noninteger ν)? • Our highly efficient “keyhole” contour of Figure 1 was discovered experimentally. What other analytical problems might be approached in this (rather unexpected) fashion? For that matter, how might one properly use the celebrated Watson loop- integral lemma with error term [22, Theorem 5.1] on our keyhole contour to obtain similar effecive asymptotics? • How can one go to arbitrary asymptotic-expansion orders with the exp-arc method in the presence of oscillatory behavior; beyond, say, Theorem 9? If this is possible for Bessel functions, it may well apply also to the more formidable Laguerre polynomials. • What is the fastest was to evaluate the converging Bessel sum (66) to, say, extreme precision? If the argument z is real, what is a good way to avoid complex arithmetic per se for the evaluation of Γ(1/2, iz)—being as we know that all coefficients bk, ck will end up real-valued?

47 9 Acknowledgements

The authors are indebted to J. Buhler, R. Mayer, O’Yeat Chan, R. Paris, and N. Temme for the generous lending of expertise, ideas and suggestions.

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